Problem 68 - Essential Calculus Early Transcendentals

Question

Two particles travel along the space curves $$\mathbf{r}_{1}(t)=\left\langle t, t^{2}, t^{3}\right\rangle \quad \mathbf{r}_{2}(t)=\langle 1+2 t, 1+6 t, 1+14 t\rangle$$ Do the particles collide? Do their paths intersect?

Step-by-Step Solution

Answer

The particles do not collide, but their paths intersect at different times.

Step 1: Determine Collision Condition

For the particles to collide, their position vectors must be equal at the same time, i.e., \( \mathbf{r}_1(t) = \mathbf{r}_2(t) \) for some \( t \).

Step 2: Set Up Equation of Position Equality

Equate the components of the two vectors: \( t = 1+2t \), \( t^2 = 1+6t \), \( t^3 = 1+14t \). These equations arise from \( \mathbf{r}_1(t) = \mathbf{r}_2(t) \).

Step 3: Solve for Simultaneous Conditions

Solve the first equation for \( t \): \( t = 1+2t \) implies \( t = -1 \).Check if this satisfies the other two equations: \( t^2 = 1+6t \), substituting \( t = -1 \) gives \( 1 = 1 - 6 \), which is false. Therefore, \( t = -1 \) is not a solution for the system.

Step 4: Determine Intersection Condition

For the paths to intersect, there must be different \( t_1 \) and \( t_2 \) such that \( \mathbf{r}_1(t_1) = \mathbf{r}_2(t_2) \). Assume \( \langle t_1, t_1^2, t_1^3 \rangle = \langle 1+2t_2, 1+6t_2, 1+14t_2 \rangle \) and solve for \( t_1 \) and \( t_2 \).

Step 5: Solve Intersection Equations

Solve the equations from Step 4, starting with the linear system: \( t_1 = 1+2t_2 \) and \( t_1^2 = 1+6t_2 \).Substitute \( t_1 = 1+2t_2 \) into the second equation: \( (1+2t_2)^2 = 1+6t_2 \), simplify this to form a quadratic in \( t_2 \).

Step 6: Verify Solutions

The quadratic equation \( 1 + 4t_2 + 4t_2^2 = 1 + 6t_2 \) simplifies to \( 4t_2^2 - 2t_2 = 0 \), which factors into \( 2t_2(2t_2-1) = 0 \). Solving gives \( t_2 = 0 \) or \( t_2 = \frac{1}{2} \).With \( t_2 = 0 \), \( t_1 = 1 \) and with \( t_2 = \frac{1}{2} \), \( t_1 = 2 \). Both lead to satisfying all three equations, confirming paths intersect.

Key Concepts

Space Curves

Space curves are fascinating mathematical structures describing the paths followed by objects in three-dimensional space. In our context, each particle's trajectory can be described using a vector function. For example, the functions for the particles given by:

• \( \mathbf{r}_{1}(t) = \langle t, t^{2}, t^{3} \rangle \)
• \( \mathbf{r}_{2}(t) = \langle 1+2t, 1+6t, 1+14t \rangle \)

These functions describe how the position of each particle changes with the parameter \( t \). Space curves can often resemble twisted paths that meander through space, and understanding intersections allows us to study interactions between different curves.

Collision Condition

A collision between two particles is quite a specific event. It occurs when both particles meet at the exact same point and at the same time. Mathematically, this is expressed as their position vectors becoming equal simultaneously:

• \( \mathbf{r}_1(t) = \mathbf{r}_2(t) \)

This condition indicates that, given specific values of \( t \), both particles are at the same location. In our exercise, solving the equations \( t = 1 + 2t \), \( t^2 = 1 + 6t \), and \( t^3 = 1 + 14t \) reveals whether such a value exists.

Intersection Condition

When we talk about intersection conditions, we are looking at a broader scenario than collision. Here, we want to find out if the paths of the particles cross at any point in space, regardless of time. This happens if there exist different times \( t_1 \) and \( t_2 \) such that the paths have the same spatial coordinates. This can be shown as:

• \( \langle t_1, t_1^2, t_1^3 \rangle = \langle 1+2t_2, 1+6t_2, 1+14t_2 \rangle \)

By solving these equations, we can determine the times \( t_1 \) and \( t_2 \) such that the paths intersect.

Position Vectors

Position vectors offer a way of describing a particle's position in space relative to an origin. For each point on the curve described by a function \( \mathbf{r}(t) \), the position vector gives the coordinates \( \langle x(t), y(t), z(t) \rangle \). This essentially tells us where the particle is at time \( t \). Understanding the role of position vectors is crucial in analyzing the movement and interaction of particles. They allow us to graphically represent motion and calculate positions over time. Position vectors in our exercise demonstrate how each particle's path and movement can be succinctly encapsulated in a mathematical form, allowing for easy manipulation and analysis.